Almost coinciding families and gaps in \({\mathcal P}(\omega)\)
type: Item
instance of: scholarly article
title: Almost coinciding families and gaps in \({\mathcal P}(\omega)\)
zbMATH Open document ID: 0799.03058
author: Shizuo Kamo
published in: Journal of the Mathematical Society of Japan
publication date: 1994-01-16T00:00:00Z
review text: The author studies nontrivial almost coinciding families indexed by \(\omega^ \omega\), abbreviated by \(F\). A. Dow et al. have shown that the existence of such an \(F\) is independent of the negation of the continuum hypothesis, CH. MA is Martin's axiom. It is shown that the following additional axioms imply the non-existence of a family \(F\): (a) If \(P\) is a poset which adjoins \(\omega_ 2\) Cohen reals, then in \(V^ P\) such an \(F\) does not exist. (b) Assume \(\omega_ 1< \kappa= \kappa^{<\kappa}\) (forcing notation). There exists a poset \(P\) with the countable chain condition such that in \(V^ P\) holds \(2^ \omega= \kappa\) + MA + there exists an unfilled \((\kappa, \kappa)\)- gap + such an \(F\) does not exist. The author poses the question whether or not ``ZFC + \(d>\omega_ 1\) + there exists such an \(F\)'' is consistent.
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E50
zbMATH DE Number: 469067
zbMATH Keywords: poset with the countable chain condition
zbMATH Keywords: forcing notation
zbMATH Keywords: almost coinciding families
zbMATH Keywords: continuum hypothesis
zbMATH Keywords: Cohen reals
reviewed by: Gudrun Kalmbach
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04520357
OpenAlex ID: W1999882760
DOI: 10.2969/JMSJ/04520357
description: scientific article; zbMATH DE number 469067